Why is $\left\{ n\in \mathbb{N}:n^2 \right\}$ nonsense, $ $ but $\left\{ n^2: n\in \mathbb{N} \right\}$ correct? Why is $\left\{ n\in \mathbb{N}:n^2   \right\}$ nonsense,  $ $ but $\left\{ n^2: n\in \mathbb{N}   \right\}$ correct?
From my understanding, $\left\{ n\in \mathbb{N}:n^2   \right\}$ should be read as:
"The set of natural numbers such that each natural number is multiplied to itself."
What's wrong with this?
Why is it not equivalent to $\left\{ n^2: n\in \mathbb{N}   \right\}$? $ $ which is read as:
"The set of $n^2$'s such that $n$ is a natural number." 
 A: In more formal notation, we'd not allow either expression.
The more formal notation in general is, given s set $X$ and a proposition $P$ of one variable:
$$\{x\in X: P(x)\}$$
So, for example, in your case:
$$\{k\in\mathbb N: \exists n\in \mathbb N(k=n^2)\}$$
Formal language is a mess sometimes, so for human readability purposes, we sometimes simplify.  For example, if the context is clear, we might write the above as:
$$\{k:\exists n (k=n^2)\}$$
That is, technically, ambiguous, but the context of the discussion might make it clear.
We can also say $\{n^2:n\in \mathbb N\}$. This is shorthand, but it is fairly direct shorthand.
The key to the notation, then, is that the left side before the $:$ separator is describing the elements. It is giving an "example" element a name, and then giving additional conditions after the $:$. But the stuff after the $:$ doesn't "change the value" on the left side of the $:$ - it is not a map, it is a filter, eliminating some of the possible values.
In English, we'd read $\{x\in X:P(x)\}$ as "The set of $x$ in $X$ such that $P(x)$ is true."
In English, we'd read $\{n^2:n\in\mathbb N\}$ as "The set of $n^2$ such that $n$ is a natural number."
Your sentence for $\{n\in\mathbb N: n^2\}$ is not in that form - it is not a "such that" expression, but a "followed by some operation applied to $n$." That is not how the notation is used.
A: All forms of this notation are of the form
$$ \{ \text{term denoting an element of the set} \mid \text{additional information} \} $$
You have the pieces the other way around, which is why it's gramatically incorrect.

There are two primary ways this notation is used; the first is to pick out the elements of some set satisfying a property. For example, the set of integer square roots of $9$ would be written as
$$ \{ n \in \mathbb{Z} \mid n^2 = 9 \} $$
The key elements are, on the left hand side, we introduce the variable $n$ which refers to an arbitrary element of the set, along with specifying its type: that $n$ is a natural number. (there are various reasons why we do this on the left hand side rather than the right)
On the right hand side, we put down the condition that $n$ must obey to be an element of the set we construct.
The other way to use this notation is by applying a transformation to elements of some set. For example, the set of all integer multiples of $2$ is
$$ \{ 2n \mid n \in \mathbb{Z} \} $$
In this usage, the variable $n$ is introduced on the right hand side, and is expressing an arbitrary element of the domain we are applying the transformation to. The left hand side denotes the element of the set we are constructing that corresponds to the element $n$ from the domain.
A: If $A$ is a set, and $\varphi$ is a function mapping an element $a \in A$ to a statement $\varphi(a)$ about $a$ that is either true or false, then we denote the set of all $a \in A$ such that the corresponding $\varphi(a)$ is true as 
$$\{ a \in A | \varphi(a) \}.$$
So in the definition, the order matters. First we see a dummy variable $a$ for the elements of $A$, and after the $|$ sign we get a statement about this $a$. 
Now if we take $A=\mathbb{N}$, then you are basically giving $\varphi$ as $\varphi(n)=n^2$. But numbers aren't statements. For example, how could we check whether $25$, which is $\varphi(5)$, is true? It is simply not a meaningful question, $25$ is not true or false. 
Now the notation $\{n^2  | n \in \mathbb{N} \}$ is another way of writing the image of the function from $\mathbb{N}$ to itself given by $n \mapsto n^2$. I have to recognize this last mentioned notation is a little bit abusive, since "the set of all $n^2$ such that $n \in \mathbb{N}$" is a an awkward sentence, because strictly speaking you don't know what set $n$ or $n^2$ is in until after the 'such that'. 
A: There should be a condition after the colon (read as "such that") which limits the set, for example $\{n \in \mathbb N : \sqrt{n} \in \mathbb N \}$ is ok.
Your example was like saying

find all $n$ where $n \times n$

Where $n \times n$ what? Is less than some value? Is equal to some value? Is an element of a set?
A: What does "The set of natural numbers such that each natural number is multiplied to itself." mean? That would mean:

"The set of natural numbers...

OK, at this point, you should have $\{1,2,3,\ldots\}$ in your imagination.

such that each natural number is multiplied to itself."

OK, now look for the numbers in the list above that are "multiplied to itself", and only keep those numbers. But I don't know what that would mean. Is $3$ "multiplied to itself"? You could do that I guess. Then all the the numbers in this list qualify, and they are all part of the set you are describing, and you end up with $\{1,2,3,\ldots\}$, not $\{1,4,9,\ldots\}$.
I think maybe you are thinking of a different English construction that would be more like "Take the set of natural numbers, then multiply each such number by itself." But that is not what set-builder notation conventionally means.
A: To confirm what everybody else has said, after the colon, you need a sentence that may or may not be true. Since there’s no verb in what follows the colon, your formulation is bad grammar.
